Multiple Regression

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Thu, 19 Nov 2009 08:45:41 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/19/t12586456162o6izwfin868nx8.htm/, Retrieved Thu, 19 Nov 2009 16:47:09 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/19/t12586456162o6izwfin868nx8.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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1.4 1.9 -0.7 -2.9 -0.8 1 1 1.6 -0.7 -0.7 -2.9 -0.8 -0.8 0 1.5 -0.7 -0.7 -2.9 -2.9 -1.3 3 1.5 -0.7 -0.7 -0.7 -0.4 3.2 3 1.5 -0.7 -0.7 -0.3 3.1 3.2 3 1.5 1.5 1.4 3.9 3.1 3.2 3 3 2.6 1 3.9 3.1 3.2 3.2 2.8 1.3 1 3.9 3.1 3.1 2.6 0.8 1.3 1 3.9 3.9 3.4 1.2 0.8 1.3 1 1 1.7 2.9 1.2 0.8 1.3 1.3 1.2 3.9 2.9 1.2 0.8 0.8 0 4.5 3.9 2.9 1.2 1.2 0 4.5 4.5 3.9 2.9 2.9 1.6 3.3 4.5 4.5 3.9 3.9 2.5 2 3.3 4.5 4.5 4.5 3.2 1.5 2 3.3 4.5 4.5 3.4 1 1.5 2 3.3 3.3 2.3 2.1 1 1.5 2 2 1.9 3 2.1 1 1.5 1.5 1.7 4 3 2.1 1 1 1.9 5.1 4 3 2.1 2.1 3.3 4.5 5.1 4 3 3 3.8 4.2 4.5 5.1 4 4 4.4 3.3 4.2 4.5 5.1 5.1 4.5 2.7 3.3 4.2 4.5 4.5 3.5 1.8 2.7 3.3 4.2 4.2 3 1.4 1.8 2.7 3.3 3.3 2.8 0.5 1.4 1.8 2.7 2.7 2.9 -0.4 0.5 1.4 1.8 1.8 2.6 0.8 -0.4 0.5 1.4 1.4 2.1 0.7 0.8 -0.4 0.5 0.5 1.5 1.9 0.7 0.8 -0.4 -0.4 1.1 2 1.9 0.7 0.8 0.8 1.5 1.1 2 1.9 0.7 0.7 1.7 0.9 1.1 2 1.9 1.9 2.3 0.4 0.9 1.1 2 2 2.3 0.7 0.4 0.9 1.1 1.1 1.9 2.1 0.7 0.4 0.9 0.9 2 2.8 2.1 0.7 0.4 0.4 1.6 3.9 2.8 2.1 0.7 0.7 1.2 3.5 3.9 2.8 2.1 2.1 1.9 etc...

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation bbp[t] = -0.658401005013655 + 0.862606433366398dnst[t] -0.0400106821199718y1[t] -0.0938026651675497y2[t] + 0.0544948659153816y3[t] + 0.389840352423138y4[t] + 0.0762801278972633M1[t] + 0.250610875389412M2[t] + 0.735839254790127M3[t] + 0.576575956377957M4[t] + 0.842257409297771M5[t] + 0.58792273935397M6[t] + 0.501342706463074M7[t] + 0.605788032914381M8[t] + 0.565962578885114M9[t] + 0.657248181394328M10[t] + 0.424207069846181M11[t] -0.00773667959424864t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) -0.658401005013655 0.469228 -1.4032 0.168098 0.084049 dnst 0.862606433366398 0.130378 6.6162 0 0 y1 -0.0400106821199718 0.112815 -0.3547 0.724665 0.362332 y2 -0.0938026651675497 0.145014 -0.6469 0.521333 0.260667 y3 0.0544948659153816 0.146718 0.3714 0.712232 0.356116 y4 0.389840352423138 0.133275 2.9251 0.005589 0.002795 M1 0.0762801278972633 0.450718 0.1692 0.866439 0.43322 M2 0.250610875389412 0.445299 0.5628 0.576641 0.288321 M3 0.735839254790127 0.451297 1.6305 0.110656 0.055328 M4 0.576575956377957 0.458038 1.2588 0.215225 0.107613 M5 0.842257409297771 0.445936 1.8887 0.066015 0.033008 M6 0.58792273935397 0.457028 1.2864 0.205519 0.102759 M7 0.501342706463074 0.455131 1.1015 0.277088 0.138544 M8 0.605788032914381 0.456644 1.3266 0.19198 0.09599 M9 0.565962578885114 0.459473 1.2318 0.225055 0.112528 M10 0.657248181394328 0.450472 1.459 0.152179 0.076089 M11 0.424207069846181 0.445081 0.9531 0.346125 0.173062 t -0.00773667959424864 0.005566 -1.3901 0.172004 0.086002 Multiple Linear Regression - Regression Statistics Multiple R 0.934025997985242 R-squared 0.872404564912327 Adjusted R-squared 0.819499140607682 F-TEST (value) 16.4898888229072 F-TEST (DF numerator) 17 F-TEST (DF denominator) 41 p-value 3.46611628287974e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.652448276746705 Sum Squared Residuals 17.4532389070196 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 1.69537433284622 -0.295374332846218 2 1 0.580668754581649 0.419331245418351 3 -0.8 -1.10880937473681 0.308809374736813 4 -2.9 -1.80593082733721 -1.09406917266279 5 -0.7 -0.800457693143368 0.100457693143368 6 -0.7 -0.0516367899623158 -0.648363210037684 7 1.5 1.89350865691398 -0.393508656913976 8 3 3.13885245371768 -0.138852453717678 9 3.2 3.52844798860734 -0.328447988607336 10 3.1 3.58517733714265 -0.485177337142647 11 3.9 2.95119319017667 0.94880680982333 12 1 1.03698395111164 -0.0369839511116440 13 1.3 0.301626739981237 0.998373260018763 14 0.8 -0.436138573574668 1.23613857357467 15 1.2 0.702294992165985 0.497705007834015 16 2.9 2.38601539806213 0.513984601937865 17 3.9 3.81878725782836 0.0812127421716392 18 4.5 4.23709537832613 0.262904621673867 19 4.5 3.84355487756027 0.656445122439733 20 3.3 2.46024613885832 0.839753861141675 21 2 1.70628127712671 0.293718722873288 22 1.5 1.35790000889298 0.142099991107018 23 1 1.62969885591364 -0.629698855913642 24 2.1 2.73157877387005 -0.631578773870052 25 3 3.74949494752284 -0.749494947522844 26 4 4.89393075701504 -0.893930757015044 27 5.1 5.31585923685243 -0.215859236852432 28 4.5 4.21254655343733 0.287453446562665 29 4.2 3.75606154484843 0.443938455151569 30 3.3 3.11204999783437 0.187950002165628 31 2.7 2.85177167769766 -0.151771677697659 32 1.8 2.52112645435853 -0.721126454358529 33 1.4 1.53379727755809 -0.133797277558085 34 0.5 0.775687310343635 -0.275687310343635 35 -0.4 0.53566161575785 -0.93566161575785 36 0.8 0.501799590911343 0.298200409088657 37 0.7 1.30854677046173 -0.60854677046173 38 1.9 2.02140922839143 -0.121409228391434 39 2 2.17204376578178 -0.172043765781784 40 1.1 1.47062995646826 -0.370629956468263 41 0.9 1.47693244797496 -0.576932447974963 42 0.4 0.953389827149578 -0.553389827149578 43 0.7 1.01077478201472 -0.310774782014717 44 2.1 2.14167762067835 -0.0416776206783466 45 2.8 2.81436721277874 -0.0143672127787354 46 3.9 2.93902496692128 0.960975033078722 47 3.5 2.55168051429118 0.948319485708824 48 2 1.62963768410696 0.370362315893035 49 2 1.34495720918797 0.655042790812029 50 1.5 2.14012983358654 -0.64012983358654 51 2.5 2.91861137993661 -0.418611379936613 52 3.1 2.43673891936948 0.663261080630523 53 2.7 2.74867644249161 -0.048676442491612 54 2.8 2.04910158665223 0.750898413347767 55 2.5 2.30039000581338 0.199609994186619 56 3 2.93809733238712 0.0619026676128784 57 3.2 3.01710624392913 0.182893756070868 58 2.8 3.14221037669946 -0.342210376699459 59 2.4 2.73176582386066 -0.331765823860662 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.177094313427606 0.354188626855212 0.822905686572394 22 0.308649953654948 0.617299907309896 0.691350046345052 23 0.747236683545128 0.505526632909743 0.252763316454872 24 0.862532660557925 0.27493467888415 0.137467339442075 25 0.941941268801892 0.116117462396216 0.0580587311981079 26 0.959401576140302 0.0811968477193967 0.0405984238596984 27 0.936746165562014 0.126507668875972 0.0632538344379858 28 0.917228239572067 0.165543520855866 0.0827717604279328 29 0.870141574612876 0.259716850774248 0.129858425387124 30 0.832117082341307 0.335765835317387 0.167882917658693 31 0.84689236206211 0.306215275875778 0.153107637937889 32 0.880593932138611 0.238812135722777 0.119406067861389 33 0.80604211218109 0.387915775637819 0.193957887818910 34 0.790084724405089 0.419830551189822 0.209915275594911 35 0.714995365288611 0.570009269422778 0.285004634711389 36 0.737155611123788 0.525688777752423 0.262844388876212 37 0.759415047154490 0.481169905691019 0.240584952845510 38 0.811573196693806 0.376853606612389 0.188426803306194 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 1 0.0555555555555556 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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